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Yin-Yang Sphere with Probability and Interactive Geometry

Yin-Yang Sphere with Probability and Interactive Geometry
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2020-03-14 23:43
FunctionSolid GeometryProbability & StatisticsParametric EquationSlider Animation
This GeoGebra example, titled 'Ying Yang', demonstrates a three-dimensional sphere decorated with the classic Yin-Yang symbol, while integrating multiple mathematical concepts such as binomial distribution, continuity, circumscribed circles, rectangles, and logarithmic functions. The core mathematical ideas involve parametric surface equations, probability distributions, geometric transformations, and function behavior. The construction uses the Surface command to generate the sphere, with objects like a, q1, q2, r, c1, c2, t1, t2 controlling shape and parameters. Interactive buttons (Bouton1, Bouton2) and a speed slider (Speed) enable dynamic animation, and an embedded image (image1) enhances the visual presentation. This resource is suitable for high school and university students studying solid geometry, probability, and parametric equations, and serves as an engaging tool for mathematical art, classroom demonstrations, or exploring the intersection of abstract math and visual design.

GeoGebra Commands & Parameters

  1. Surface(((cos(v) + cos((u * v))) * cos(a)) - (sin(a) * (sin(v) + sin((u * v)))), ((cos(v) + cos((u * v))) * sin(a)) + (cos(a) * (sin(v) + sin((u * v)))), 0, u, 0, 1, v, 0, π) → q1
  2. Surface((-(((cos(v) + cos((u * v))) * cos(a)) - (sin(a) * (sin(v) + sin((u * v)))))), 0, ((cos(v) + cos((u * v))) * sin(a)) + (cos(a) * (sin(v) + sin((u * v)))), u, 0, 1, v, 0, π) → q2
  3. Surface((((v * cos(u)) + 1) * cos(a)) - ((sin(a) * v) * sin(u)), (((v * cos(u)) + 1) * sin(a)) + ((cos(a) * v) * sin(u)), 0, u, 0, (2 * π), v, 0, r) → c1
  4. Surface((-((((v * cos(u)) + 1) * cos(a)) - ((sin(a) * v) * sin(u)))), 0, (((v * cos(u)) + 1) * sin(a)) + ((cos(a) * v) * sin(u)), u, 0, (2 * π), v, 0, r) → c2
  5. Surface((((v * cos(u)) - 1) * cos(a)) - ((sin(a) * v) * sin(u)), (((v * cos(u)) - 1) * sin(a)) + ((cos(a) * v) * sin(u)), 0, u, 0, (2 * π), v, r, 1) → t1
  6. Surface((-((((v * cos(u)) - 1) * cos(a)) - ((sin(a) * v) * sin(u)))), 0, (((v * cos(u)) - 1) * sin(a)) + ((cos(a) * v) * sin(u)), u, 0, (2 * π), v, r, 1) → t2
  7. Surface(cos(a) + (((0.9 * r) * cos(u)) * cos(v)), sin(a) + (((0.9 * r) * cos(u)) * sin(v)), ((0.9 * r) * sin(u)), u, 0, (2 * π), v, 0, (2 * π)) → b
  8. Surface((-cos(a)) + (((0.9 * r) * cos(u)) * cos(v)), (((0.9 * r) * cos(u)) * sin(v)), sin(a) + ((0.9 * r) * sin(u)), u, 0, (2 * π), v, 0, (2 * π)) → c